The GL(1|1) WZW model: From Supergeometry to Logarithmic CFT

TL;DR

This paper provides a complete solution to the GL(1|1) WZW model, revealing its spectrum, indecomposable representations, and logarithmic correlators, and explores its representation theory and spectral flow symmetry.

Contribution

It offers a detailed analysis of the GL(1|1) WZW model, including its spectrum, indecomposable representations, and correlation functions, connecting supergeometry to logarithmic conformal field theory.

Findings

Spectrum contains indecomposable representations

Correlation functions exhibit logarithmic behavior

Model has spectral flow symmetry

Abstract

We present a complete solution of the WZW model on the supergroup GL(1|1). Our analysis begins with a careful study of its minisuperspace limit (``harmonic analysis on the supergroup''). Its spectrum is shown to contain indecomposable representations. This is interpreted as a geometric signal for the appearance of logarithms in the correlators of the full field theory. We then discuss the representation theory of the gl(1|1) current algebra and propose an Ansatz for the state space of the WZW model. The latter is established through an explicit computation of the correlation function. We show in particular, that the 4-point functions of the theory factorize on the proposed set of states and that the model possesses an interesting spectral flow symmetry. The note concludes with some remarks on generalizations to other supergroups.